SOLUTION: please help me with this....Find the​ vertical, horizontal, and oblique​ asymptotes, if​ any, for the given rational function. Q(x)=(3x^2-11x-4)/(2x^2-7x-4)

Algebra ->  Rational-functions -> SOLUTION: please help me with this....Find the​ vertical, horizontal, and oblique​ asymptotes, if​ any, for the given rational function. Q(x)=(3x^2-11x-4)/(2x^2-7x-4)       Log On


   

Question 1157588: please help me with this....Find the​ vertical, horizontal, and oblique​ asymptotes, if​ any, for the given rational function.
Q(x)=(3x^2-11x-4)/(2x^2-7x-4)
Found 2 solutions by MathLover1, greenestamps:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
Q%28x%29=%283x%5E2-11x-4%29%2F%282x%5E2-7x-4%29
Horizontal asymptote:
find quotient of highest degree of variable
3x%5E2%2F2x%5E2=3%2F2
%283x%5E2-11x-4%29%2F%282x%5E2-7x-4%29->3%2F2 as x-> ±infinity

Vertical asymptote:
find values excluded from domain
2x%5E2-7x-4=0
%28x-4%29%282x%2B1%29=0
x=4
2x=-1->x=-1%2F2

%283x%5E2-11x-4%29%2F%282x%5E2-7x-4%29-> ±infinity as x->-1%2F2
%283x%5E2-11x-4%29%2F%282x%5E2-7x-4%29-> undefined as x->4
=> Vertical asymptote: x=-1%2F2
oblique​ asymptote:
since numerator and denominator have same highest degree of variable x, there is no+oblique asymptotes



Answer by greenestamps(13203) About Me  (Show Source):
You can put this solution on YOUR website!


(1) Horizontal or oblique asymptotes

The degrees of the numerator and denominator are the same, so there is a horizontal asymptote and no oblique asymptote.

The ratio of the leading coefficients gives you the horizontal asymptote.

ANSWER: Horizontal asymptote at y = 3/2.

(2) Vertical asymptotes

Vertical asymptotes will occur wherever there is a linear factor in the denominator that is not also in the numerator.

Factor numerator and denominator:

%28%283x%2B1%29%28x-4%29%29%2F%28%282x%2B1%29%28x-4%29%29

There is a vertical asymptote were the factor (2x+1) in the denominator is equal to 0.

ANSWER: There is a single vertical asymptote, at x = -1/2.

What about the factors (x-4) in both numerator and denominator?

For x=4, the denominator is 0 and so the function is undefined. For all other values of x, the function is equivalent to %283x%2B1%29%2F%282x%2B1%29. So the graph of the given function is the same as the graph of 3x%2B1%29%2F%282x%2B1%29 except that there is a hole in the graph at x=4.